Nuprl Lemma : free_abmon_unique
∀S:DSet. ∀M,N:FAbMon(S). ∃f:MonHom(M.mon,N.mon). ∃g:MonHom(N.mon,M.mon). InvFuns(|M.mon|;|N.mon|;f;g)
Proof
Definitions occuring in Statement :
free_abmon_mon: f.mon,
free_abmonoid: FAbMon(S),
inv_funs: InvFuns(A;B;f;g),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
monoid_hom: MonHom(M1,M2),
grp_car: |g|,
dset: DSet
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
subtype_rel: A ⊆r B,
inv_funs: InvFuns(A;B;f;g),
and: P ∧ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
unique_set: {!x:T | P[x]},
monoid_hom: MonHom(M1,M2),
so_lambda: λ2x.t[x],
so_apply: x[s],
dset: DSet,
implies: P ⇒ Q,
squash: ↓T,
abmonoid: AbMon,
mon: Mon,
true: True,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
free_abmonoid_wf,
dset_wf,
free_abmon_umap_wf,
free_abmon_mon_wf,
free_abmon_inj_wf,
inv_funs_wf,
grp_car_wf,
unique_set_wf,
equal_wf,
compose_wf,
exists_wf,
monoid_hom_wf,
free_abmon_endomorph_is_id,
compose_wf_for_mon_hom,
set_car_wf,
squash_wf,
true_wf,
abmonoid_wf,
comp_assoc,
iff_weakening_equal,
free_abmon_umap_properties_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
hypothesis,
dependent_pairFormation,
applyEquality,
because_Cache,
sqequalRule,
independent_pairFormation,
isectElimination,
lambdaEquality,
setElimination,
rename,
functionEquality,
independent_functionElimination,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
cumulativity
Latex:
\mforall{}S:DSet. \mforall{}M,N:FAbMon(S).
\mexists{}f:MonHom(M.mon,N.mon). \mexists{}g:MonHom(N.mon,M.mon). InvFuns(|M.mon|;|N.mon|;f;g)
Date html generated:
2017_10_01-AM-10_01_11
Last ObjectModification:
2017_03_03-PM-01_03_42
Theory : polynom_1
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